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Stochastic Differential Equations

  • Ettore Vitali
  • Mario Motta
  • Davide Emilio Galli
Chapter
Part of the UNITEXT for Physics book series (UNITEXTPH)

Abstract

In this chapter we introduce the formalism of stochastic differential equations (SDE). After an introduction stressing their importance as generalizations of ordinary differential equations (ODE), we discuss existence and uniqueness of their solutions and we prove the Markov property. This leads us to a deep connection with the theory of partial differential equations (PDE), which will emerge naturally when computing time derivatives of averages of the processes. In particular we will introduce the generalized heat equation, as well as the more general Feynman-Kac equation, underlying the path integral formalism and the Schrödinger equation in imaginary time. Moreover, studying the time evolution of the transition probability of the processes will lead us to the Kolmogorov equations. A special case is provided by the Liouville equation, the cornerstone of classical statistical mechanics.

Keywords

Stochastic differential equations Chapman-Kolmogorov equation Fokker-Planck equation Geometric brownian motion Brownian bridge Feynman-Kac equation Kakutani representation 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ettore Vitali
    • 1
  • Mario Motta
    • 2
  • Davide Emilio Galli
    • 3
  1. 1.Department of PhysicsCollege of William and MaryWilliamsburgUSA
  2. 2.Division of Chemistry and Chemical EngineeringCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of PhysicsUniversity of MilanMilanItaly

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